<- Return to Excel Statistics Page

Global Warming and Leadership

Keith Greiner

August 11, 2020

This essay is about global warming. The essay begins with a few comments on leadership with examples of what can happen when leaders ignore the warnings of experts. It continues with a discussion of global warning related data with examples of several of my investigations of published data on carbon dioxide (CO₂) levels and, temperature levels. This leads to a regression analysis of CO₂, as an independent variable vs. temperature as a dependent variable. The data confirm reports that CO₂ is associated with increasing temperatures.

One premise of the analysis and the essay is that an average person who has studied high school or college statistics should be able to replicate the conclusions proposed by scientists who have determined that the earth is warming to a dangerous level. To that end, it uses data that are publicly available, and includes links to the data sources. In order for the analyses to work, the data must be available. Efforts by the Trump administration to suppress critical data and information are of concern and are highlighted. The data analysis tools are found on most every computer in the Microsoft Office suite. The connection to management concepts reflects fundamentals of management that are provided in almost every introductory management text.

The essay refers to two terms “Climate Change” and “Global Warming”. I have used each where it is appropriate. However, know that the most appropriate term is “Global Warming” because that term most accurately describes what is happening throughout the world. Below is a list of the topics discussed:

1. The Incident of 1707

2. Same Leadership in 2017

3. The First Global Warming Model

4. Significant Contribution of 1975 – Call it Global Warming

5. CO₂ Cycles andTrends

6. CO₂ Distribution by Latitude

7. Increasing Temperatures by U. S. State

8. Historical Worldwide Temperature Increases

9. Relationship Between CO₂ and Temperature Anomalies

10. What the Data Show

11. Post Script -- Returning to the Subject of Leadership in this Crisis

12. Additional Reading

13. 13. Appendix -- Using Scatter Charts in an Analysis

We begin by looking at an incident that occurred in 1707. It is about naval history, but I assure you, it will be related to today’s discussion of global warming.

1. The Incident of 1707

On October 22, 1707, over 1,500 British sailors lost their lives in one of the more senseless tragedies in British naval history. Commander in Chief, Admiral Sir Cloudesley Shovell was in the lead ship and the first of four to founder on the rocks off the island of Scilly (See latitude and longitude 49.868484, -6.441418 just copy and paste these latitude and longitude values into Google Maps). According to one legend, a sailor knew the area, knew the danger ahead, and warned Shovell that a course change was needed. Shovell, an arrogant know-it-all commander, not only disregarded the sailor’s expertise, but also had the un-named man hanged for mutiny. Had the sailor lived only a few more minutes, he could have offered an “I told you so”, but then, after such a disaster, it’s too late for either a statement of “I was right”, or of “I’m sorry”. When it’s over, there is nothing left but to bury the dead and move on.

Today, 310 years later, we can’t verify the legend, but we can understand the haughtiness of a so-called “leader” whose arrogance takes thousands of people on a disastrous, life-ending path. Sadly, we are facing a similar situation, today, and the Scilly legend is a good metaphoric lesson that is helpful, in this hour of worldwide crisis.

2. Same Leadership Attitude in 2017

Today, we have thousands of knowledgeable scientists (à la mode de, the sailor), warning of an impending global climate disaster, while the U. S. Commander in Chief, President Donald Trump (à la mode de, Shovell). Trump and others berate the warnings and the scientists as he sails the country into the rocks of increased carbon emissions and other greenhouse gasses that contribute to global warming. He could solve the problem, but he refuses: preferring ignorance over informed policy-making. On June 1, 2017, Trump announced that the U. S. would not participate in the global effort to reduce greenhouse gasses. That leaves the U. S. alone with Nicaragua and war-torn Syria as the only countries not participating in the Paris Accord on Climate Change. Even rogue nations of North Korea and Iran are participating. (à la mode de and it’s abbreviation à la comes from a French source and means “in the manner of…”)

Trump’s position was revealed well before his election when on November 6, 2012 he tweeted, “The concept of global warming was created by and for the Chinese in order to make U. S. manufacturing non-competitive.” (See Politifact at http://www.politifact.com/truth-o-meter/statements/2016/jun/03/hillary-clinton/yes-donald-trump-did-call-climate-change-chinese-h/ ) When Trump finally got the opportunity to affect policy, his actions confirmed that false statement.

Shortly after the January 20, 2017 inauguration, Trump nominated (and the Senate Republicans confirmed) Scott Pruitt as EPA administrator. Pruitt has a long history of suing the EPA on behalf of oil industry interests, and apparently claimed that the EPA should be abolished. It all appears to be an effort to suppress regulations that are desperately needed to help save the environment. After Pruitt became head of the agency, climate change data were removed from the agency web site. Below is the page at epa.gov/climatechange as of February 17, 2017 and again on August 15, 2017.

And below is the EPA news release on the subject.

Text of the page is indicative of the failing leadership in this regard. The page says,

“Thank you for your interest in this topic. We are currently updating our website to reflect EPA's priorities under the leadership of President Trump and Administrator Pruitt. If you're looking for an archived version of this page, you can find it on the January 19 snapshot.”

If a user clicks on the link to the January 19 snapshot, the page includes the following warning.

“This is not the current EPA website. To navigate to the current EPA website, please go to www.epa.gov. This website is historical material reflecting the EPA website as it existed on January 19, 2017. This website is no longer updated and links to external websites and some internal pages may not work.”

Fortunately, the New York Times has posted the 2017 vision of the US Global Change Research Report, Climate Science Special Report (CSSR), and the document is available on the NYT's non-government site. As of August 12, 2017 you can read the New York Times report and download the report at this address.

3. The First Global Warming Model

The first “sailor” on the worldwide environmental ship was Svante Arrhenius (1859-1927), a Swedish chemist whose 1896 paper, "On the Influence of Carbonic Acid in the Air upon the Temperature of the Ground" describes the relationship between carbonic acid (a molecule of carbon dioxide and water) and the temperature of the earth. Arrhenius suggested that the earth was in equilibrium under the generally accepted principles described below. His term “all authors” refers to others upon whom his literature review was based. Arrhenius wrote,

“All authors agree in the view that there prevails an equilibrium in the temperature of the earth and of its atmosphere. The atmosphere must, therefore, radiate as much heat to space as it gains partly through the absorption of the sun's rays, partly through the radiation from the hotter surface of the earth and by means of ascending currents of air heated by contact with the ground. On the other hand, the earth loses just as much heat by radiation o space and to the atmosphere as it gains by absorption of the sun's rays.” (See http://web.lemoyne.edu/%7Egiunta/arrhenius.html)

Given that the world was thought to be in a state of equilibrium, Arrhenius raised the question as to what might occur if that equilibrium were no longer applicable. The research question was, what would happen if the amount of carbonic acid increased? He also answered the question by providing results of calculations for temperature changes. In his words,

“… I have calculated the mean alteration of temperature that would follow if the quantity of carbonic acid varied from its present mean value (K=1) to another, viz. to K=0.67, 1.5, 2, 2.5, and 3 respectively. This calculation is made for every tenth parallel, and separately for the four seasons of the year. The variation is given in Table VII."

Arrhenius’ calculations apply Stefan’s Law of Radiation, and show that an increase in carbonic acid results in an increase in temperature. It’s that simple. The theory behind the model is described at: https://www.acs.org/content/acs/en/climatescience/atmosphericwarming/singlelayermodel.html. One way of thinking of it is this: the light that comes from the sun warms the earth. When that light arrives it has a broad spectrum of frequencies and it warms the earth. The earth, in turn, emits energy at warm infra-red frequencies. We can’t directly see the infra-red, but we can measure it with infra-red detection devices and we can feel it in the warmth of heat from, say a concrete sidewalk, driveway, or roadway. The problem is, that when CO₂ and carbonic acid, H₂CO₃ are in the atmosphere, that infra-red emission cannot escape, and is reflected back to earth or is absorbed in the atmosphere, making the earth warmer. As the proportion of CO₂ and H₂CO₃ increases, the equilibrium is reduced and the retained heat increases. Other greenhouse gases have similar effects.

4. Significant Contribution of 1975 – Call it Global Warming

The world received many warnings after Arrhenius, with a notable estimate from Wallace Broecker in August, 1975. Broecker’s article in Science Magazine revisits the relationship between CO₂ and temperature. You may obtain a copy of the article from the American Association for the Advancement Science at http://science.sciencemag.org/content/189/4201/460. The article is credited with being the origin of the term “Global Warming” and is surprisingly predictive of the situation we see today. In Broecker’s words:

“This warming would by the year 2000 bring average global temperatures beyond the range experienced during the last 1,000 years. Until chemical fuel consumption is dramatically reduced, global temperatures would continue to rise. Future natural cycles would merely modulate this ever-steepening rise.”

Below is a selection of data from Broecker’s table predicting CO₂ and temperature using an historical series of 10 year increments from 1900 to 1970 and a predicted series from 1980 to 2010.

When the CO₂ values are compared to temperature, using an Excel scatter chart, the trend line can be calculated. Graph 1 shows all values in the table. The historical values, as of 1975 are presented in blue and the estimated values are presented in red. We see that the historical data used by Broecker show a relationship that can be described by a straight line with the equation y = 0.0098 - 2.8818. That is reasonable to expect. When we examine a small segment of a series with increasing slope, it very well may look like a straight line. Broecker proposed that the slope of the line would become increasingly steeper and that estimate is shown in Graph 2, where it is compared to actual data.

This essay uses several National Oceanic and Atmospheric Administration (NOAA) sources of CO₂ data. One series begins in 1958. By aligning that series along side the data from Broecker’s paper, we have the following graph. That looks to me like Broecker’s estimates were remarkably close to the actual values reported by NOAA for the years from 1958 through 2010.

5. CO₂ Trends

As mentioned in Section 4, the National Oceanic and Atmospheric Administration (NOAA) published CO₂ data covering the period from March, 1958 through April, 2017 at ftp://aftp.cmdl.noaa.gov/products/trends/co2/co2_mm_mlo.txt as of June 3, 2017. You may be able to obtain updates there. However, I am concerned that the Trump administration might remove the data after a new director of the NOAA is appointed. That would be tragic.

Graph 3 shows a historical trend in CO₂ between March, 1958 and December, 2015. On the x axis, notice that that months are represented by their proportional part of the year. In the equation for y, the x values are sequential integers beginning with 1 in March, 1958 and incrementing by 1 for each month in the data set. That’s how Excel determines the x values in the graphics tools. CO₂ is expressed as a mole fraction in dry air, micromol/mol, abbreviated as parts per million (ppm). The line is consistently rising with has annual cyclic components.

Notice the ripples in the data. Lets take a close up look at one of the ripples. Graph 4 zooms in to one of those components and shows the consistent rise and fall of CO₂ during the months of 1959. Other years have the same pattern.

The two graphs show that although there is a consistent annual cycle, there is also an overall trend that is increasing. The trend may be described with the equation of y = 309.05e^0.0004x where x is a sequential number beginning with 1 in March 1958 and incrementing by 1 for each month. The x value for June, 2017 is 712. The trend is a smoothed version of the detailed data that is made up of smaller annual cycles.

The 12-month straight-line slope of the line from March, 1958 to March, 1959 is 1.49. The equation for that is very simple. It is the 1959 value less the 1958 value. When the equation is repeated for subsequent years, the slope of the annual change increases. The increased slopes suggest that if that overall trend in the blue line is extended, the slope of the annual change will continue to increase. In ten years, that is by March 2027, the annual rate of increase might be 2.06, and the value for March, 2017 might be about 430.

6. CO₂ Distribution by Latitude

Below is a composite of two images that NOAA presents on the pages at https://www.esrl.noaa.gov/gmd/dv/iadv/graph.php?code=WBI&program=ccgg&type=tw with the selection of “Latitude Distribution (multiple sites)”. The graphs show valuable information and I hope the page will continue to be available after the Trump administration has its way with climate-related sites. For the image shown below, I combined an image of the graph for January 1981 with the image for January 2016. The data points show CO₂ in units of micro-mol of CO₂ per mol of air. The x-axis is presented as the sine of the latitude of measurements with 1.0 (North Pole) on the right and -1.0 (South Pole) on the left. Zero represents the equator. We see that the January, 1981 measurements are all in the range of 340 with a slight increase in the northern latitudes. The January, 2016 values are between 400 and 420 with an increase in the northern latitudes. The orange values are considered preliminary until a rigorous data review process is completed. Given the difference between the two lines, and the pattern of increasing values for each month’s data (not shown here), I expect the actual January 2016 values to closely align with the preliminary values.

7. Increasing Temperatures by U. S. State

As the CO₂ and H₂CO₃ values increase, so does the temperature. While Arrhenius’ estimates were based on Stefan's Law of Radiation, scientists now have much more sophisticated measurement and analytical tools, and much more data. Today, they can see the real relationship. Following are a few of my analyses of publicly available data. Map 1 and Excel Image 1, describe the increases in U. S. temperatures, by state, between 1980 and 2017. The data come from the NOAA site https://www.ncdc.noaa.gov/temp-and-precip/state-temps/ as of June, 2017. At that address historical information for individual states can be obtained. For this project, the data were downloaded for individual states, then stored, and aggregated in Excel. What was needed for this project was a way to standardize the data and to account for the many monthly variability’s that occurred between 1980 and 2017. The method that is shown, below, accomplishes that requirement. The comparison is based on the slope of a linear regression of the monthly average temperatures, by state, that were published by NOAA. The slope was calculated in Excel using the =SLOPE( Yrange, Xrange) function. The slope is the variable “m” in the formula for a straight line in y = mx +b where the combination of m and b define the line that has the minimum deviation from all points in the time series. In this analysis, we do not need to know the value of “b”, because our only interest is the rate of change as in “m”. Linear regressions and slope calculations are the stuff of high school and college statistics classes. I add that, to suggest that these calculations are very much within the scope and abilities of anyone who has had the opportunity to take introductory statistics. If you took statistics and don't recall these items, you are not alone, and this is an opportunity to refresh your memory.

Below is a table of slopes of the January temperatures between 1980 and 2017. Color codes show markings of each state on Map 1. Remember that these are linear slopes that are calculated to represent a series of 37 years. That doesn’t mean that, if extended, they will predict future temperatures. The analysis of Graph 3 suggests that overall slope of the CO₂ line is increasing, and the temperature is likely to follow that trend.

Map 1 and Excel Image 2 show the historical growth pattern of U. S. states during the month of January. They are an application of the slopes. I used the slope of a linear regression to indicate the increase in temperature over the time from 1980 through 2017.

8. Historical Worldwide Temperature Increases

Now, lets explore the temperature growth on a worldwide basis. NOAA publishes aggregated temperature anomalies for worldwide land and ocean surface temperatures. On June 14, 2017, these data could be found at: https://www.ncdc.noaa.gov/data-access/marineocean-data/noaa-global-surface-temperature-noaaglobaltemp.

Anomalies are positive or negative values that are the difference between a current temperature and some reference point in the past. The reference point might be a single point, but more often is an average. The use of anomalies is a form of standardization that allows the comparison of values from a variety of sources. The anomalies used here are based on the climatology from 1971 to 2000.

Graph 6 shows the historical worldwide land temperature anomalies for the period from 1958 through 2015. That is the same time period of the CO₂ data shown in Graph 3. Here, we see monthly averages, and a trend line having the formula y = 3E-06x² + 0.0004x - 0.4492, where x is an integer series from 1 through 696. The black trend line shows a regression series based on the trend line formula. As with the monthly averages for CO₂, the regression trend line eliminates the variations in the chart and reveals the general trend. This is the same procedure shown in Graph1 and the analyses summarized in Excel Image 2. Graph 6 shows us that the average worldwide land temperature anomalies (and therefore the temperature) increased in the period from 1958 through 2015. The increasing slope of the trend line suggests that the rate of temperature changes is increasing.

9. Relationship Between CO₂ and Temperature Anomalies

A scatter chart may be used to reveal the relationship between two data sets having a one-for-one match between the two sets. When the CO₂ time series is compared with temperature anomalies in a scatter chart, we have Graph 7 shown below. The linear regression on the two data sets gives a value of y = 0.0176x -6.0887 where CO₂ is the independent variable and temperature anomalies are the dependent variable. Here, we see a positive relationship. The R² value of 0.29738 is low because there are many outliers and many variations from the trend line. Like the data in Graph 6, the Graph 7 scatter chart shows a very active series with a consistent increasing relationship. The Excel graphing system will not allow the addition of an exponential trend line, so we're not able to see that. The exponential trend line appears in the scatter chart shown in Graph 8.

Because we also smoothed the two data sets for CO₂ and for temperature anomalies, the two can be brought together In a scatter chart in Graph 8. Here, we see the relationship between the two is so close that it forms a single line. The correlation between the two series is 0.98 and the R² value is 0.97. The formula for the trend line is definitely positive, with a slight curve as in y = 0.0001x² – 0.0702 + 9.51153. On one hand, that is to be expected, given that the smoothed lines have similar upward trends. On another hand, the chart indicates a relationship between the two series, and that relationship is shown to be positive.

10. What the Data Show

This analysis is certainly not as thorough as the many studies reported by climatologists and chemists who devote their entire careers to research on the subject of climate change. I am more an analyst than an expert on global warming. My view is that there is a lot of publicly available data that an average person can use to reveal similar patterns and to confirm the findings of experts. In many cases, the data should be the subject of statistical analysis exercises for students in college and high school statics classes.

By applying some very basic, generally accepted analyses and by using the ubiquitous Microsoft Office data analysis tools with publicly available data from credible sources, I have come to a point that suggests the same conclusion as all those other studies. The earth temperature is increasing at an ever-increasing rate, and there is a positive relationship between the smoothed CO₂ values and the smoothed temperature anomaly values. If we were to extend the trend lines in Graphs 5 and 6 to the right then we would be able to estimate that as the independent variable CO₂ increases, we can expect that the worldwide land temperature would also increase. The models, that I have described here, provide different answers, and that’s ok. This is a stochastic thing, and variability is to be expected and a ballpark estimate is the best that can be obtained, by me, in these circumstances. The worldwide effect of temperature increases on plants and animals is beyond the scope of this essay, and one should consult other sources for that. What I know is that, in recent history, temperature increases are already significant and, many would say, catastrophic.

11. Returning to the Subject of Leadership in this Crisis

There are about as many perspectives on management and leadership as there are people on earth. From a functional perspective, most of those perspectives can be reduced to the four functions that have come from late 20th century and early 21st century rendition of Henri Fayol's functions: planning, organizing, leading, and controlling. In this case, "controlling" is a single word for the process of gathering, analyzing, and responding to feedback. While the responsibility for "controlling" activity may be assigned to certain people in an organization, when it comes to global warming, in a democracy, the "controlling" part of management is a responsibility of everyone who has some say, large or small, in the administration of our worldwide environment. Just as all of us manages the temperature of our homes, so we also have a moral responsibility with regard to the temperature of the planet.

It's about managing the processes that affect worldwide temperature. As a way of comparison, I will use examples of another important community resource: water quality. Let's say, for example, that the manager/leader is running a water treatment plant. The processing of controlling the process involves a periodic chemical analysis of the chemicals in the water, evaluating results of the analysis and adjusting the process to ensure that all residents receive the highest standards of water quality. The process of controlling requires that incoming data be accurate and reliable. It requires that the report be truthful. And that the reporter be believed.

In 2015, pennlive.com reported that a Steeltown treatment plant operator was charged with with two counts of tampering with public records or information and two counts of penalties and remedies related to the Safe Drinking Water Act" (see http://www.pennlive.com/midstate/index.ssf/2015/06/steelton_water_authority_chief.html )

And, of course, there is the problem with the Flint, Michigan water supply (see http://www.cnn.com/2016/03/04/us/flint-water-crisis-fast-facts/index.html). The Flint system became highly contaminated after he water source was switched to the polluted Flint River. Officials received accurate reports of problems and failed to act. Hundreds were affected. Lives were affected. Many levels of administrative and elected leadership failed to respond and on June 14, 2017 the New York Times reported that five were charged with manslaughter ( see https://mobile.nytimes.com/2017/06/14/us/flint-water-crisis-manslaughter.html?_r=0&referer=https://www.google.com/ ).

The differences between these examples is in the degree of impact. In the Steeltown case, the report makes no mention of injuries. While the reporting failed, there was an oversight "control" agency available to catch it and take corrective action. In the case of Flint, reports were made, but the oversight agency, the leadership, failed to take action. No action, is in fact, action. And in the Flint case, lives were lost and people were permanently harmed.

It seems to me that the case of Global Warming is many many times worse than Flint. It is many times worse than our opening example fom the 1707 incident off the southern British coast. Global Warming will impact everyone on earth. Some of the predictions are dire, and the obvious fact that our national leadership is ignoring it, will eventually be seen as a dereliction of duty.

At the very minimum, our leaders should think of Global Warming as a threat that requires a contingency plan. That would be something like the contingency plans that organizations develop for the purpose of having a plan in place if there is a catastrophic event, with the hope that no such event ever occurs. In past years, such contingency planning has been an essential part of every State government. What we're seeing at the Federal level suggests that the very idea of contingency planning is not in the Trump administration's limited vocabulary. That also could be seen as a dereliction of duty.

The data show Global Warming is highly likely and extremely dangerous. Scientists all over the world are calling for action. Political leaders should not take the warnings lightly, and should take immediate action.

12. Additional Reading

Here are some sources for additional reading.

1. Intergovernmental Panel on Climate Change: http://www.ipcc.c

2. Washington University: http://cses.washington.edu/db/pdf/snoveretalsok2013appendix.pdf

3. American Chemical Society: https://www.acs.org/content/acs/en/policy/publicpolicies/sustainability/globalclimatechange.html

4. American Chemical Society Toolkit: https://www.acs.org/content/acs/en/climatescience.html

5. American Meteorological Society: https://www.ametsoc.org/ams/index.cfm/about-ams/ams-statements/statements-of-the-ams-in-force/climate-change/

6. Wallace S. Breocker Essay: http://www.funnyordie.com/articles/44d2ccb862/the-scientist-who-named-it-global-warming-would-like-to-apologize

7. Wallace S. Breocker Scientific Paper, 1975 Climatic Change: Are We on the Brink of a Pronounced Global Warming? Author(s): Wallace S. Broecker Source: Science, New Series, Vol. 189, No. 4201 (Aug. 8, 1975), pp. 460-463 Published by: American Association for the Advancement of Science Stable URL: http://www.jstor.org/stable/1740491

   1. US Global Change Research Report, Climate Science Special Report (CSSR) Final Draft -- 2017, https://www.nytimes.com/interactive/2017/08/07/climate/document-Draft-of-the-Climate-Science-Special-Report.html.

   2. Statement from Iowa academic climate and related scientists: 2017

13. Appendix -- Using Scatter Charts in an Analysis

This essay features three scatter graphs (See Graphs 1, 7, and 8). For those who don’t use scatter graphs every day, this section will explain some of the basic ideas. Microsoft Excel calls them Scatter Charts. Others may use “scatter grams” (or “scattergrams”), or “scatter plots”, or some similar term. No matter what they are called, they are plots of two or more variables’ points onto the same graph space. This discussion will use two variables.

The two matched variables are plotted on the same chart with variable x (independent variable) on the x-axis, and variable y (dependent variable) on the y-axis. So the first step is to have two matched variables. In the case of global warming, we have a collection of records, where each record has a number for CO₂ and a number for temperature where all values represent a specific time. There may be annual averages, monthly average, weekly averages, daily averages or hourly averages, etc. Graph 1 uses annual averages, whle Graphs 7 and 8 use monthly averages. Having matched records is important because without the match there is no meaning to the scatter chart. Step 2 is to align the two sets of matched records in an Excel spreadsheet with the independent variable on the left and the dependent variable on the right. Put the appropriate headings at the top.

How do you know which is the independent and which is the dependent variable? That is where your knowledge of the subject matters. In the case of carbon dioxide in the atmosphere, countless chemists have investigated what happens when the molecule carbon dioxide interacts with other elements and molecules. Further, from an understanding of the carbon and oxygen and hydrogen atoms, they are able to determine what happens when CO₂ and carbonic acid, H₂CO₃, interact with light.

When the graph is to be used for a regression analysis, the selection of independent vs. dependent variables suggests causality: that is, which variable causes the other to occur? Causality requires human judgment. One example is the causality of turning on a switch vs. lighting a light. We know that turning a switch causes a light to light because we can reach up and turn the switch (that makes it an independent variable), and then the lights become bright. Of course, electricity must be present, but in the space of existing electricity, proper connections and the act of turning the switch, we know the turn of the switch causes the light to become bright. If the switch is a variable switch (ie: a dimmer dial), we can plot the position of the dial and measure the intensity of light, and that can be plotted on a scatter chart.

On that basis, we can start the CO₂ and temperature analysis with carbon dioxide as the independent variable. Therefore, CO₂ goes into the left-hand column, while temperature goes into the right-hand column. Now, we go through the steps to create a scatter chart in Excel. Excel gives several options for scatter charts with some having connected lines. For this analysis you want to use the option with no lines. In my Excel, that is the first choice. Often some further adjustment is needed. I often need to adjust the upper and lower scale limits to spread the chart throughout the display space. With the display selected, right-click on one of the dots and set the options. I often add a trend line, with the option to show the equation and the R² value. You will see trend lines and equations in all of Graphs, 1, 7, and 8. The default is a linear equation, and Excel offers several other useful options.

Below are several variations on the appearance of scatter charts. The titles describe the contents of the charts. Notice how the R² values change, based on the dispersion of the y values in charts.

--

--

---

---

---

-30-

<- Return to Excel Statistics Page